It's unusual for the overflow opening of a bathroom sink to get clogged, given that the sink is seldom filled to the point at which its services are required. When it does get blocked, the problem could be an accumulation of hair or a small object that somehow got forced through one of the overflow holes. Because the overflow passage is small and follows the contours of the sink, you can't treat it like a regular drain, but you still have at least three options for unblocking it.
Use a Zip-It Tool
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The Zip-It tool is one of those simple plumbing inventions, like a plunger, that no household should be without. It's thin and flexible like a cable tie, so it fits into virtually any drain, and its barbed construction makes it perfect for hooking onto hair and dragging it out of the drain. You could substitute a pipe cleaner, as suggested by Merry Maids, but a Zip-It tool is longer, grabs hair more efficiently and is inexpensive, costing about $5.
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A Zip-It tool is about 12 inches long, which is long enough to reach all the way through the overflow passage to the point at which it empties into the drain. Insert it into the overflow opening, push it in as far as it will go and extract it. If the drain is clogged by hair or anything similarly sinuous, it should come out with the tool.
Use a Plunger
The plunger is the first choice of most plumbers for clearing a clogged drain, and although it seems counterintuitive, you can use one to clear a sink overflow passage. You don't do this by putting the plunger over the overflow opening. That would be futile, since a plunger can't make a seal against the side of the sink. Instead, you plunge the drain and take advantage of the fact that the overflow passage is connected to it.
Fill the sink about half full of water, tilt the plunger into the water to fill the cup, then place the cup over the sink drain and start plunging while you open the stopper. The pumping action will force anything in the overflow passage to back up, and you'll know you're successful when water (and possibly gunk) come shooting out of the overflow opening.
Some Disassembly May Be Required
If the blockage is caused by a small, hard object, you may not be able to force it backward with a plunger, and you can't hook it with a Zip-It tool. Caustic drain cleaning chemicals won't work either, and you should avoid these anyway, because they can harm your pipes. That leaves you one last option, which is to disassemble the drain and manually remove it.
Start by disconnecting the pop-up stopper and removing the stopper lever from the drain tailpiece. It's possible that the object got lodged on the lever, and if so, that's all the disassembly you need to do. It may help to push the Zip-It tool into the lever inlet and probe until you contact the object and are able to dislodge it.
If all else fails, unscrew the nut on the bottom of the sink holding the tailpiece, using adjustable joint pliers and push the tailpiece out of the sink. This exposes the overflow passage outlet in the drain opening and allows you to extract the object with a screwdriver or a pair of needle-nose pliers. After extracting the obstruction, don't forget to repack the tailpiece flange with fresh plumber's putty before you reassemble the drain.